A trapezoid is a four-sided figure that contains one pair of

32. In the rectangle shown, what is true of the lengths of each

pair of opposite sides? R S V T

33. A line segment is bisected if its two parts have the same

length. Which line segment, or , is bisected at point X? CD AB C A B X 5 cm 3 cm 3 cm 3 cm D

34. An angle is bisected if its two parts have the same

measure. Use three letters to name the angle that is bisected. 15 10 10 B E D C A In Exercises 35 to 38, where A-B-C on , it follows that . AB + BC = AC AC

35. Find AC if AB

= 9 and BC = 13.

36. Find AB if AC

= 25 and BC = 11.

37. Find x if AB

= x , BC = x + 3, and AC = 21.

38. Find an expression for AC the length of if AB

= x and BC = y .

39. ⬔ABC is a straight angle. Using your protractor, you can

show that m ⬔1 + m ⬔2 = 180°. Find m ⬔1 if m⬔2 = 56°. AC C A B Exercises 35–38 A B 1 2 C D Exercises 39, 40

40. Find m

⬔1 if m⬔1 = 2x and m ⬔2 = x . HINT: See Exercise 39. In Exercises 41 to 44, m ⬔1 + m ⬔2 = m ⬔ABC.

41. Find m ⬔ABC if m⬔1

= 32° and m ⬔2 = 39°.

42. Find m ⬔1 if m⬔ABC

= 68° and m ⬔1 = m ⬔2.

43. Find x if m ⬔1

= x , m ⬔2 = 2x + 3, and m ⬔ABC = 72°.

44. Find an expression for m ⬔ABC if m⬔1

= x and m ⬔2 = y . 1 C B D A 2 Exercises 41–44

45. A compass was used to mark off three congruent

segments, , , and . Thus, has been trisected at points B and C. If AD = 32.7, how long is ? AB AD CD BC AB B C D A E

46. Use your compass and straightedge to bisect .

EF F E

47. In the figure, m ⬔1

= x and m ⬔2 = y . If x - y = 24°, find x and y. HINT: m⬔1 + m⬔2 = 180 ⬚.

48. In the drawing, m

⬔1 = x and m ⬔2 = y . If m ⬔RSV = 67° and x - y = 17°, find x and y. HINT: m⬔1 + m⬔2 = m⬔RSV. R S T V 1 2 D C A B 1 2 For Exercises 49 and 50, use the following information. Relative to its point of departure or some other point of reference, the angle that is used to locate the position of a ship or airplane is called its bearing. The bearing may also be used to describe the direction in which the airplane or ship is moving. By using an angle between 0° and 90°, a bearing is measured from the North-South line toward the East or West. In the diagram, airplane A which is 250 miles from Chicago’s O’Hare airport’s control tower has a bearing of S 53° W.

49. Find the bearing of airplane B relative to the control

tower.

50. Find the bearing of airplane C relative to the control

tower. 53 250 mi 325 mi 300 mi 24 22 control tower W E S N B C A Exercises 49, 50 A MATHEMATICAL SYSTEM Like algebra, the branch of mathematics called geometry is a mathematical system. Each system has its own vocabulary and properties. In the formal study of a mathemat- ical system, we begin with undefined terms. Building on this foundation, we can then define additional terms. Once the terminology is sufficiently developed, certain prop- erties characteristics of the system become apparent. These properties are known as axioms or postulates of the system; more generally, such statements are called as- sumptions. Once we have developed a vocabulary and accepted certain postulates, many principles follow logically when we apply deductive methods. These statements can be proved and are called theorems. The following box summarizes the components of a mathematical system sometimes called a logical system or deductive system. Mathematical System Axiom or Postulate Theorem Ruler Postulate Distance Segment-Addition Postulate Midpoint of a Line Segment Ray Opposite Rays Intersection of Two Geometric Figures Plane Coplanar Points Space Early Definitions and Postulates 1.3 KEY CONCEPTS FOUR PARTS OF A MATHEMATICAL SYSTEM 1. Undefined terms

2. Defined terms

f vocabulary

3. Axioms or postulates 4. Theorems

f principles